Chapter 5 Genetic Algorithmshomes.di.unimi.it/.../Didattica/AlgoritmiEuristici/MaterialeAE/Ch05-GA… · Genetic Algorithms Colin R. Reeves Abstract Genetic algorithms (GAs) have

Chapter 5Genetic Algorithms

Colin R. Reeves

Abstract Genetic algorithms (GAs) have become popular as a means of solvinghard combinatorial optimization problems. The first part of this chapter brieflytraces their history, explains the basic concepts and discusses some of their the-oretical aspects. It also references a number of sources for further research intotheir applications. The second part concentrates on the detailed implementation ofa GA. It discusses the fundamentals of encoding a ‘genotype’ in different circum-stances and describes the mechanics of population selection and management andthe choice of genetic ‘operators’ for generating new populations. In closing, somespecific guidelines for using GAs in practice are provided.

5.1 Introduction

The term genetic algorithm, almost universally abbreviated nowadays to GA, wasfirst used by John Holland [1], whose book Adaptation in Natural and ArtificialSystems of 1975 was instrumental in creating what is now a flourishing field of re-search and application that goes much wider than the original GA. Many people nowuse the term evolutionary computing or evolutionary algorithms (EAs), in order tocover the developments of the last 15 years. However, in the context of metaheuris-tics, it is probably fair to say that GAs in their original form encapsulate most ofwhat one needs to know.

Holland’s influence in the development of the topic has been very important, butseveral other scientists with different backgrounds were also involved in develop-ing similar ideas. In 1960s Germany, Ingo Rechenberg [2] and Hans-Paul Schwefel[3] developed the idea of the Evolutionsstrategie (in English, evolution strategy),while—also in the 1960s—Bremermann, Fogel and others in the USA implemented

Colin R. ReevesDepartment of Mathematics, Statistics and Engineering Science, Coventry University, Priory St,Coventry, UKe-mail: [email protected]

M. Gendreau, J.-Y. Potvin (eds.), Handbook of Metaheuristics, 109International Series in Operations Research & Management Science 146,DOI 10.1007/978-1-4419-1665-5 5, c© Springer Science+Business Media, LLC 2010

110 Colin R. Reeves

their idea for what they called evolutionary programming. The common thread inthese ideas was the use of mutation and selection—the concepts at the core ofthe neo-Darwinian theory of evolution.1 Although some promising results wereobtained, evolutionary computing did not really take off until the 1980s. Note theleast important reason for this was that the techniques needed a great deal of compu-tational power. Nevertheless, the work of these early pioneers is fascinating to readin the light of our current knowledge; David Fogel (son of one of the early pioneers)has documented some of this work in [4].

1975 was a pivotal year in the development of genetic algorithms. It was inthat year that Holland’s book was published, but perhaps more relevantly for thoseinterested in metaheuristics, that year also saw the completion of a PhD thesis byone of Holland’s graduate students, Ken De Jong [5]. Other students of Holland’shad completed theses in this area before, but this was the first to provide a thoroughtreatment of the GA’s capabilities in optimization.

A series of further studies followed, the first conference on the nascent subjectwas convened in 1985, and another graduate student of Holland’s, David Goldberg,produced first an award-winning PhD thesis on his application to gas pipeline opti-mization, and then, in 1989, an influential book [6]—Genetic Algorithms in Search,Optimization, and Machine Learning. This was the final catalyst in setting off asustained development of GA theory and applications that is still growing rapidly.

Optimization has a fairly small place in Holland’s work on adaptive systems, yetthe majority of research on GAs tends to assume this is their purpose. De Jong,who initiated this interest in optimization, has cautioned that this emphasis maybe misplaced in a paper [7] in which he contends that GAs are not really functionoptimizers, and that this is in some ways incidental to the main theme of adaptation.Nevertheless, using GAs for optimization is very popular, and frequently successfulin real applications, and to those interested in metaheuristics, it will undoubtedly bethe viewpoint that is most useful.

Unlike the earlier evolutionary algorithms, which focused on mutation and couldbe considered as straightforward developments of hill-climbing methods, Holland’sGA had an extra ingredient—the idea of recombination. It is interesting in this re-gard to compare some of the ideas being put forward in the 1960s in the field ofoperational research (OR).

OR workers had by that time begun to develop techniques that seemed able toprovide ‘good’ solutions, even if the quality was not provably optimal (or even near-optimal). Such methods became known as heuristics. A popular technique, whichremains at the heart of many of the metaheuristics described in this handbook, wasthat of neighbourhood search, which has been used to attack a vast range of com-binatorial optimization problems. The basic idea is to explore ‘neighbours’ of anexisting solution—these being defined as solutions obtainable by a specified opera-tion on the base solution.

One of the most influential papers in this context was that published by Lin [8],who found excellent solutions to the travelling salesman problem by investigating

1 Well-meaning attempts to read off the validity or otherwise of Darwinism from the performanceof GAs are illegitimate. GAs are clear examples of ‘intelligent design’.

5 Genetic Algorithms 111

neighbourhoods formed by breaking any three links of a tour and re-connectingthem. Empirically, Lin found that these ‘3-optimal’ solutions were of excellentquality—in the case of the (rather small) problems he investigated, often close tothe global optimum. However, he also made another interesting observation andsuggested a way of exploiting it. While starting with different initial permutationsgave different 3-optimal solutions, these 3-optimal solutions were observed to havea lot of features (links) in common. Lin therefore suggested that search should beconcentrated on those links about which there was not a consensus, leaving the com-mon characteristics of the solutions alone. This was not a GA as Holland was de-veloping it, but there are clear resonances. Much later, after GAs had become morewidely known, Lin’s ideas were re-discovered as ‘multi-parent recombination’ and‘consensus operators’.

Other OR research of the same era took up these ideas. Roberts and Flores [9](apparently independently) used a similar approach to Lin’s for the TSP, whileNugent et al. [10] applied this basic idea for the quadratic assignment problem.However, the general principle was not adopted into OR methodology, and rela-tively little was done to exploit the idea until GAs came on the OR scene in the1990s.

In what follows, Section 5.2 provides an overview of the basic GA concepts,Section 5.3 gives a sketch of the theoretical background, while Section 5.4 listssome important sources for further exploration. The remaining sections focus onthe various stages required for the implementation of a GA.

5.2 Basic Concepts

Assume we have a discrete search space X and a function

f : X �→ IR.

The general problem is to findarg min

x∈Xf .

Here, x is a vector of decision variables, and f is the objective function. We assumehere that the problem is one of minimization, but the modifications necessary fora maximization problem are nearly always obvious. Such a problem is commonlycalled a discrete or combinatorial optimization problem (COP).

One of the distinctive features of the GA approach is to allow the separation ofthe representation of the problem from the actual variables in which it was originallyformulated. In line with biological usage of the terms, it has become customary todistinguish the ‘genotype’—the encoded representation of the variables, from the‘phenotype’—the set of variables themselves. That is, the vector x is representedby a string s, of length l, made up of symbols drawn from an alphabet A , using amapping

c : A l �→ X .

112 Colin R. Reeves

In practice, we may need to use a search space

S ⊆A l ,

to reflect the fact that some strings in the image of A l under c may represent invalidsolutions to the original problem. (This is a potential source of difficulty for GAsin combinatorial optimization—a topic that is covered in [11].) The string lengthl depends on the dimensions of both X and A , and the elements of the stringcorrespond to ‘genes’, and the values those genes can take to ‘alleles’. This is of-ten designated as the genotype–phenotype mapping. Thus the optimization problembecomes one of finding

arg mins∈S

g,

where the functiong(s) = f (c(s)).

It is usually desirable that c should be a bijection. (The important property of abijection is that it has an inverse, i.e., there is a unique vector x for every string s,and a unique string s for every vector x.) In some cases the nature of this mappingitself creates difficulties for a GA in solving optimization problems.

In using this device, Holland’s ideas are clearly distinct from the similar method-ology developed by Rechenberg [2] and Schwefel [3], who preferred to work withthe original decision variables directly. Both Holland’s and Goldberg’s books claimthat representing the variables by binary strings (i.e., A = {0,1}) is in some sense‘optimal’, and although this idea has been challenged, it is still often convenientfrom a mathematical standpoint to consider the binary case. Certainly, much of thetheoretical work in GAs tends to make this assumption. In applications, many rep-resentations are possible—some of the alternatives that can be used in particularCOPs are discussed in [11].

The original motivation for the GA approach was a biological analogy. In theselective breeding of plants or animals, for example, offspring are sought that havecertain desirable characteristics—characteristics that are determined at the geneticlevel by the way the parents’ chromosomes combine. In the case of GAs, a popula-tion of strings is used, and these strings are often referred to in the GA literature aschromosomes. The recombination of strings is carried out using simple analogies ofgenetic crossover and mutation, and the search is guided by the results of evaluatingthe objective function f for each string in the population. Based on this evaluation,strings that have higher fitness (i.e., represent better solutions) can be identified, andthese are given more opportunity to breed. It is also relevant to point out here thatfitness is not necessarily to be identified simply with the composition f (c(s)); moregenerally, fitness is h( f (c(s))) where h : IR �→ IR+ is a suitable monotonic functionused to eliminate the problem of ‘negative’ fitness.

Perhaps the most fundamental characteristic of genetic algorithms is their useof populations of many strings. Here again, the German ‘evolution strategy’ (ES)school initially did not use populations and focused almost exclusively on ‘muta-tion’ operators which are generally closer in concept to the types of operator used


in neighbourhood search and its extensions. Holland did use mutation, but in hisscheme it is generally treated as subordinate to crossover. Thus, in Holland’s GA,instead of the search moving from point to point as in methods based on localsearch, the whole set of strings undergoes ‘reproduction’ in order to generate a newpopulation.

De Jong’s work established that population-based GAs using crossover and mu-tation operators could successfully deal with optimization problems of several dif-ferent types, and in the years since his work was published, the application of GAsto COPs has grown almost exponentially.

These operators and some developments of them are described more fully inSections 5.9 and 5.10. At this point, however, it might be helpful to provide a verybasic introduction. Crossover is a matter of replacing some of the genes in one parentby corresponding genes of the other. An example of one-point crossover would bethe following. Given the parents P1 and P2, with crossover point 3 (indicated byX), the offspring will be the pair O1 and O2:

P1 1 0 1 0 0 1 0 O1 1 0 1 1 0 0 1X

P2 0 1 1 1 0 0 1 O2 0 1 1 0 0 1 0

The other common operator is mutation in which a gene (or subset of genes) ischosen randomly and the allele value of the chosen genes is changed. In the caseof binary strings, this simply means complementing the chosen bit(s). For example,the string O1 above, with genes 3 and 5 mutated, would become 1 0 0 1 1 0 1.A simple template for the operation of a genetic algorithm is shown in Figure 5.1.The individual parts of this very general formulation will be discussed in detail later.

Choose an initial population of chromosomes;while termination condition not satisfied do

repeatif crossover condition satisfied then{select parent chromosomes;choose crossover parameters;perform crossover};if mutation condition satisfied then{choose mutation points;perform mutation};evaluate fitness of offspring

until sufficient offspring created;select new population;endwhile

Fig. 5.1 A genetic algorithm template. This is a fairly general formulation, accommodating manydifferent forms of selection, crossover and mutation. It assumes user-specified conditions underwhich crossover and mutation are performed, a new population is created, and whereby the wholeprocess is terminated.

114 Colin R. Reeves

5.3 Why Does It Work?

Exactly how and why GAs work is still hotly debated. There are various schoolsof thought, and none can be said to provide a definitive answer. A comprehensivesurvey is available in [12]. Meanwhile, the following is a brief guide to the mainconcepts that have been used.

5.3.1 The ‘Traditional’ View

Holland’s explanation of why it is advantageous to search the space A l rather thanX hinges on three main ideas. Central to this understanding is the concept of aschema (plural schemata). A schema is a subset of the space A l in which all thestrings share a particular set of defined values. This can be represented by using thealphabet A ∪∗ ; in the binary case, 1 * * 1, for example, represents the subset ofthe 4-dimensional hypercube {0,1}4 in which both the first and the last genes takethe value 1, i.e., the strings {1 0 0 1, 1 0 1 1, 1 1 0 1, 1 1 1 1}.

The first of Holland’s ideas is that of intrinsic (also known as implicit)parallelism—the notion that information on many schemata can be processed inparallel. Under certain conditions that depend on population size and schema char-acteristics, Holland estimated that a population of size M contains information onO(M3) schemata. However, these schemata cannot actually be processed in parallel,because independent estimates of their fitness cannot be obtained in general [14].

The second concept is expressed by the so-called Schema Theorem, in whichHolland showed that if there are N(S, t) instances of schema S in the population attime t, then at the next time step (following reproduction), the expected number ofinstances in the new population can be bounded by

E[N(S, t +1)]≥ F(S, t)F(t)

N(S, t){1− ε(S, t)},

where F(S, t) is the fitness of schema S, F(t) is the average fitness of the popula-tion, and ε(S, t) is a term that reflects the potential for genetic operators to destroyinstances of schema S.

By failing to appreciate the stochastic and dynamic nature of this relationship,somewhat extravagant conclusions have been drawn from this theorem, expressedin the frequently made statement that good schemata will receive exponentially in-creasing numbers of trials in subsequent generations. However, it is clear that theSchema Theorem is a result in expectation only, and even then for just one genera-tion. Any attempt to extrapolate this result for more than one generation is doomedto failure because the terms are then no longer independent of what is happening inthe rest of the population. Moreover, given the finite population size, it is clear thatany exponential increase cannot last very long.

Holland also attempted to model schema processing (or hyperplane competi-tions) by means of an analogy to stochastic two-armed bandit problems. This is


a well-known statistical problem: we are given two ‘levers’ which if pulled give‘payoff’ values according to different probability distributions. The problem is touse the results of previous pulls in order to maximize the overall future expectedpayoff. In [1] it is argued that a GA approximates an ‘optimal’ strategy which al-locates an (exponentially) increasing number of trials to the observed better lever;this is then used to contend for the supposed efficiency of a GA in distinguishingbetween competing schemata and hyperplanes.

Early accounts of GAs suggested quite strongly that in a GA we had thus dis-covered an algorithm that used the best available search strategy to solve not merelyone, but many hyperplane competitions at once: the ‘only case where combinatorialexplosion works in our favour’. Unfortunately, Wolpert and Macready’s ‘No-Free-Lunch’ Theorem (NFLT) [13] has rather destroyed such dreams.2

In fact, intrinsic parallelism turns out to be of strictly limited application; itmerely describes the number of schemata that are likely to be present in some num-bers given certain assumptions about string length, population size and (most impor-tantly) the way in which the population has been generated—and the last assumptionis unlikely to be true except at a very early stage of the search. Even then, only invery unusual circumstances—that of orthogonal populations [14]—could the hy-perplane competitions actually be processed in parallel; normally, the competitionsare not independent. The two-armed bandit analogy also fails in at least two ways:Macready and Wolpert [15] have firstly argued that there is no reason to believe thatthe strategy described by Holland as approximated by a GA is an optimal one, whilethey also believe there is also a flaw in Holland’s mathematics.

This is not to say that the Schema Theorem in particular, or the idea of a schemain general, is useless, but that what it says is of limited and mainly short-termvalue—principally, that certain schemata are likely to increase their presence in thenext population, and that those schemata will be on the average fitter, and less re-sistant to destruction by crossover and mutation, than those that do not. Neverthe-less, several researchers are working on new ways of formulating and understandingschema theory, while connecting it to other approaches; a recent summary can befound in [16].

This brings us to the third assumption implicit in the implementation of a GA—that the recombination of small pieces of the genotype (good schemata) into big-ger pieces is indeed a sensible method of finding optimal solutions. Goldberg [6]calls this the building-block hypothesis (BBH). There is certainly some negativeevidence, in that problems constructed to contain misleading building blocks mayindeed be hard for a GA to solve. The failure of the BBH is often invoked as anexplanation when a GA fails to solve particular COPs.

However, the properties of these problems are not usually such that they areuniquely difficult for GAs. Holland himself, with two other co-workers, lookedfor positive evidence in favour of the building-block hypothesis [17] and found

2 The NFLT, put simply, says that on the average, nothing—ant colonies, GAs, simulated an-nealing, tabu search, etc.—is better than random search. Success comes from adapting the tech-nique to the problem at hand, which of course implies some input of information from theresearcher.

116 Colin R. Reeves

the results rather problematical: functions constructed precisely to provide a ‘royalroad’ made up of building blocks of increasing size and fitness turned out to bemuch more efficiently solved by ‘non-genetic’ methods.

5.3.2 Other Approaches

By writing his theorem in the form of a lower bound, Holland was able to make astatement about schema S that is independent of what happens to other schemata.However, in practice what happens to schema S will influence the survival (or oth-erwise) of other schemata, and what happens to other schemata will affect whathappens to S, as is made plain by the exact models of Vose [18] and Whitley [19].

Markov chain theory [18, 19] has been applied to GAs [20–22] to gain a betterunderstanding of the GA as a whole. However, while the results are fascinating inilluminating some nuances of GA behaviour, the computational requirements areformidable for all but the smallest of problems, as shown by De Jong et al. [22], forexample.

Shapiro et al. [23] first examined GAs from a statistical mechanics perspective,and there is a growing literature on this topic. Peck and Dhawan [24] have linkedGAs to global randomized search methods. But one of the difficulties in analyzingGAs is that there is not a single generic GA, the behaviour of which will charac-terize the class of algorithms that it represents. In practice, there is a vast num-ber of ways of implementing a GA, as will be seen in the discussion later, andwhat works in one case may not work in another. Some workers have thereforetried to look for ways of predicting algorithm performance for particular problemclasses.

Reeves and Wright [14] summarize a perspective based on relating GAs to statis-tical methods of experimental design, which draws upon the biological concept ofepistasis. This expresses the idea that the expression of a chromosome is not merelya sum of the effects of its individual alleles, but that the alleles located in somegenes influence the expression of the alleles in others. From a mathematical view-point, epistasis is equivalent to the existence of interactions in the fitness function. Ifwe knew the extent of these non-linearities, we might be able to choose an appropri-ate algorithm. Unfortunately, as is explained in [25], it is unlikely that this approachwill be successful, although the literature surrounding the question of epistasis hasproduced some useful insights into GAs.

Several authors [26–28] have pointed out connections between GAs and neigh-bourhood search methods, and this has led to a considerable literature on the anal-ysis of problem landscapes. The concept of a landscape has been used informallyfor many years, but recent work [29, 30] has put the idea on a rigorous mathemat-ical foundation which is still being explored. Some of its uses in the context ofGAs is described in [31]. It appears that this way of thinking about algorithms hasgreat potential for unifying different metaheuristics and increasing our understand-ing of them.


5.4 Applications and Sources

There are numerous examples of the successful application of GAs to combinatorialoptimization problems. Books such as those by Davis [32] and Chambers [33, 34]are useful in displaying the range of problems to which GAs have been applied.In a chapter such as this, it is impossible to give an exhaustive survey of relevantapplications of GAs, but [11] lists some of the more useful and accessible refer-ences that should be of interest to people who are experimenting with metaheuris-tics. However, because of the enormous growth in reported applications of GAs, thislist is inevitably incomplete, as well as somewhat dated already. For a time, Alanderattempted to maintain a comprehensive bibliography: an early version of this is in-cluded in [34]. However, this is one area where the phenomenon of exponentialgrowth is indubitable, and the sheer number of papers published in the last 15 yearshave rather overwhelmed this enterprise. Nonetheless, updates are made availableperiodically of selected papers in specific areas—the one of most interest to readersof this book being the OR bibliography [35], which is claimed to be comprehensiveup to 1998, although it also includes some papers published later.

For more information on applications, and on GAs in general, the reader hasseveral useful books to choose from: the early ones by Holland, Goldberg andMichalewicz [1, 6, 36] tend to be over-committed to the schema-processing pointof view, but they are all still useful sources of information. Reeves [37] also reflectsthe state of the theory at the time the book was written, although it covers otherheuristic methods too. More recently, Mitchell [38] and Falkenauer [39] demon-strate a more careful approach to schemata, and Back [40] covers the wider field ofevolutionary algorithms. Eiben and Smith [41] also provide an elementary overviewof the whole field, while—in contrast—Spears [42] offers an in-depth study on thetrade-off between mutation and crossover.

All are worth consulting, but the best book now available is the recent workby De Jong [43]. For a very rigorous theoretical study, there is the book by Vose[44], which deals mainly with the Markov chain and dynamical systems approach,while Reeves and Rowe [12] have surveyed in some detail several other theoreticalperspectives on GAs. Another rigorous theoretical study is that by Schmitt [45, 46].

There are now also many conferences on GAs and related topics—too many tolist in detail. The original biennial International Conference on Genetic Algorithmsseries [47–53] is still of considerable historical interest3, while the IEEE establishedan alternative series under the title of the Congress on Evolutionary Computation[54–56]. These have now merged, under the auspices of the ACM since 2005, tocreate the annual GECCO series of conferences [57–59]. In Europe, there are twobiennial series of somewhat wider scope: the Parallel Problem Solving from Natureseries [67–72] and the International Conference on Artificial Neural Networks andGenetic Algorithms [77–82], recently renamed the International Conference onAdaptive and Natural Computing Algorithms [83, 84]. For the theoretically minded,

3 Apart from the intrinsic interest of these papers, it is well worth checking to see if someone hastried your bright new idea already!

118 Colin R. Reeves

there is a biennial workshop to consider—the Foundations of Genetic Algorithmsseries [85–93].

There are also many journals now publishing GA-related research. The majorGA journals are Evolutionary Computation (MIT Press) and IEEE Transactionson Evolutionary Computation (IEEE); other theoretical articles appear in journalsrelated to AI or to complex systems. Most OR journals—INFORMS Journal onComputing, Computers and OR, Journal of the OR Society, European Journal ofOR, etc.—have frequent papers on GAs, mainly applications. There are discussiongroups on the Internet (comp.ai.genetic), and the moderated news digest [email protected].

5.5 Initial Population

The previous sections have provided an overview of the underlying concepts, butit should be clear already that implementation of a GA requires many practicaldecisions. The major initial questions to consider relate to the population: first itssize and second the method by which its individuals are chosen. The size of thepopulation has been approached from several theoretical points of view, althoughthe underlying idea is always of a trade-off between efficiency and effectiveness.Intuitively, it would seem that there should be some ‘optimal’ value for a givenstring length, on the grounds that too small a population would not allow sufficientroom for exploring the search space effectively, while too large a population wouldso impair the efficiency of the method that no solution could be expected in a reason-able amount of time. Goldberg [94, 95] was probably the first to attempt to answerthis question, using the idea of schemata. Unfortunately, from this viewpoint, it ap-peared that the population size M should increase as an exponential function of thestring length. Experimental evidence [96, 97] suggests that populations of the sizeproposed by Goldberg’s theory are not necessary.

A slightly different question that we could ask is regarding a minimum populationsize for a meaningful search to take place. In Reeves [98], the initial principle wasadopted that, at the very least, every point in the search space should be reachablefrom the initial population by crossover only. This requirement can only be satisfiedif there is at least one instance of every allele at each locus in the whole populationof strings. On the assumption that the initial population is generated by a randomsample with replacement (which is a conservative assumption in this context), theprobability that at least one allele is present at each locus can be found. For binarystrings this is easily seen to be

P∗2 = (1− (1/2)M−1)l ,

from which we can calculate that, for example, a population of size 17 is enough toensure that the required probability exceeds 99.9% for strings of length 50. For q-aryalphabets, the calculation is somewhat less straightforward, but expressions are


given in [98] that can be converted numerically into graphs for specified confidencelevels. The results of this work suggested that a population growth of O(log l) wouldbe sufficient to cover the search space.

Finally, as to how the population is chosen, it is nearly always assumed that ini-tialization should be random. Rees and Koehler [99], using a model-based approachthat draws on the theoretical work of Vose [18], have demonstrated that samplingwithout replacement is preferable in the context of very small populations. Moregenerally, it is obvious that randomly chosen points do not necessarily cover thesearch space uniformly, and there may be advantages in terms of coverage if weuse more sophisticated statistical methods, especially for non-binary alphabets. Onesuch simple idea is a generalization of the Latin hypercube which can be illustratedas follows:

Suppose each gene has 5 alleles, labelled 0, . . . ,4. We choose the populationsize M to be a multiple of 5, and the alleles in each ‘column’ are generated as anindependent random permutation of 0, . . . ,(M− 1), which is then taken modulo 5.Figure 5.2 shows an example for a population of size 10. To obtain search spacecoverage at this level with simple random initialization would need a much largerpopulation.

Individual Gene1 0 1 3 0 2 42 1 4 4 2 3 03 0 0 1 2 4 34 2 4 0 3 1 45 3 3 0 4 4 26 4 1 2 4 3 07 2 0 1 3 0 18 1 3 3 1 2 29 4 2 2 1 1 310 3 2 4 0 0 1

Fig. 5.2 An example of Latin hypercube sampling for l = 6 and |A | = 5. Notice that each alleleoccurs exactly twice for each gene.

Another point to mention here is the possibility of ‘seeding’ the initial popula-tion with known good solutions. Some reports (e.g., in [100, 101]) have found thatincluding a high-quality solution, obtained from another heuristic technique, canhelp a GA find better solutions rather more quickly than it can from a random start.However, there is also the possibility of inducing premature convergence [102, 103].

5.6 Termination

Unlike simple neighbourhood search methods that terminate when a local optimumis reached, GAs are stochastic search methods that could in principle run for ever.In practice, a termination criterion is needed; common approaches are to set a limit

120 Colin R. Reeves

on the number of fitness evaluations or the computer clock time or to track thepopulation’s diversity and stop when this falls below a preset threshold. The mean-ing of diversity in the latter case is not always obvious, and it could relate eitherto the genotype or to the phenotype, or even, conceivably, to the fitnesses, but themost common way to measure it is by genotype statistics. For example, we coulddecide to terminate a run if at every locus the proportion of one particular allele roseabove 90%. Some attempts have been made to attack this problem from a theoreticalpoint of view [104, 105], but as they are based on the idea of finding a probabilis-tic guarantee that all possible strings have been seen, their practical application islimited.

5.7 Crossover Condition

Given the stress on recombination in Holland’s original work, it might be thoughtthat crossover should always be used, but in fact there is no reason to suppose that ithas to be so. Thus, while we could follow a strategy of crossover-AND-mutation togenerate new offspring, it is also possible to use crossover-OR-mutation. There aremany examples of both in the literature. The first strategy initially tries to carry outcrossover, then attempts mutation on the offspring (either or both). It is conceivablethat in some cases nothing actually happens at all with this strategy—the offspringare simply clones of the parents. Others always do something, either crossover ormutation, but not both. (Even then, cloning is still possible with crossover if theparents are too alike.)

The mechanism for implementing such choices is customarily a randomized rule,whereby the operation is carried out if a pseudo-random uniform deviate exceeds athreshold value. In the case of crossover, this is often called the crossover rate, oftendenoted by the symbol χ . For mutation, we have a choice between describing thenumber of mutations per string or per bit; bit-wise mutation, at a rate denoted by μ ,is more common.

In the -OR- case, there is a further possibility of modifying the relative propor-tions of crossover and mutation as the search progresses. Davis [32] has argued thatdifferent rates are appropriate at different times: high crossover at the start, highmutation as the population converges. In fact, he has suggested that the operatorproportions could be adapted online, in accordance with their track record in find-ing new high-quality chromosomes.

5.8 Selection

The basic idea of selection is that it should be related to fitness, and the originalscheme for its implementation is commonly known as the roulette-wheel method.It uses a probability distribution for selection in which the selection probability of


a given string is proportional to its fitness. Figure 5.3 provides a simple example ofroulette-wheel selection (RWS). Pseudo-random numbers are used one at a time tochoose strings for parenthood. For example, in Figure 5.3, the number 0.13 wouldselect string 1, the number 0.68 would select string 4.

0.00

0.32

0.410.58

0.75

0.00

0.32

0.410.58

0.751

23

4

51

23

4

5

Fig. 5.3 Suppose there are five strings in a population with fitnesses {32,9,17,17,25}, respec-tively. The probability of selection of each individual is proportional to the area of a sector of aroulette-wheel (or equivalently, to the angle subtended at the centre). The numbers on the spokesof the wheel are the cumulative probabilities for use by a pseudo-random number generator. Onthe left we have standard roulette-wheel selection, with a single pointer that has to be spun fivetimes. On the right we have SUS, using five connected equally spaced pointers; one spin providesfive selections.

Finding the appropriate number for a given pseudo-random number r requiressearching an array for values that bracket r—this can be done in O(logM) time fora population of size M. However, this approach has a high stochastic variability, andthe actual number of times NC that chromosome C is selected in any generation maybe very different from its expected value E[NC]. For this reason, sampling withoutreplacement may be used to ensure that at least the integral part of E[NC] is achieved,with fractions being allocated using random sampling.

In practice, Baker’s stochastic universal selection (SUS) [106] is a particularlyeffective way of realizing this outcome. Instead of a single choice at each stage, weimagine that the roulette wheel has an equally spaced multi-armed spinner. Spinningthe wheel produces simultaneously the values NC for all the chromosomes in thepopulation. From the viewpoint of statistical sampling theory, this corresponds tosystematic sampling [107]. Experimental work by Hancock [108] clearly demon-strates the superiority of this approach, although much published work on applica-tions of GAs still appears to rely on the basic roulette-wheel method4.

An associated problem is that of finding a suitable measure of fitness for themembers of the population. Simply using the objective function values f (x) is rarelysufficient, because the scale on which f (x) is measured is important. (For example,values of 10 and 20 are much more clearly distinguished than 1010 and 1020.)Also, in some cases, observed values of f may be negative, which complicates

4 Note that the purpose of SUS is not to reduce the total of random numbers needed. Havinggenerated a multiset of size M as our ‘mating pool’, we still have to decide which pairs matetogether, whereas in RWS we can simply pair them in the order generated.

122 Colin R. Reeves

fitness-proportional schemes. Further, if the objective is minimization rather thanmaximization, a transformation is clearly required.

Some sort of scaling is thus usually applied, and Goldberg [6] gives a simplealgorithm to deal with both minimization and maximization. The method is cum-bersome, however, and it needs continual re-scaling as the search progresses. Twoalternatives provide more elegant solutions.

5.8.1 Ranking

Ranking the chromosomes in fitness order loses some information, but there is noneed for re-scaling, and selection algorithm is simpler and more efficient. Supposethe probability of selecting the string that is ranked kth in the population is denotedby P[k]. In the case of linear ranking, we assume that

P[k] = α +βk,

where α and β are constants. The requirement that P[k] be a probability distributiongives us one condition:

M

∑k=1

(α +βk) = 1,

which leaves us free to choose the other parameter in a way that tunes the selectionpressure. This term is loosely used in many papers and articles on GAs. Here, wemean the following:

Definition 5.1 Selection pressure

φ =Prob.[selecting fittest string]

Prob.[selecting average string].

In the case of linear ranking, we interpret the average as meaning the median string,so that

φ =α +βM

α +β (M +1)/2

(This assumes the population size is odd—however, the analysis holds mutatismutandis for the case of an even number.) Some simple algebra soon establishesthat

β =2(φ −1)

M(M−1)and α =

2M−φ(M +1)M(M−1)

which implies that 1 ≤ φ ≤ 2. With this framework, it is easy to see that thecumulative probability distribution can be stated in terms of the sum of an arithmeticprogression, so that finding the appropriate k for a given pseudo-random number ris simply a matter of solving the quadratic equation


α k +βk(k +1)

2= r,

for k, which can be done simply in O(1) time. The formula is

k =−(2α +β )±

√(2α +β )2 +4β r

2β.

In contrast, searching for k (given a value for r) using ordinary fitness-proportionalselection needs at least O(logM) time.

Other functions can be used besides linear ranking [108, 109] but the abovescheme is sufficiently flexible for most applications.

5.8.2 Tournament Selection

The other alternative to strict fitness-proportional selection is tournament selection,in which a set of τ chromosomes are chosen and compared, the best one beingselected for parenthood. This approach has similar properties to linear ranking forτ = 2. It is easy to see that the best string will be selected every time it is compared,while the median string will be chosen with probability 2−(τ−1). Thus the selectionpressure is given by φ = 2τ−1, which for τ = 2 is similar to linear ranking whenα → 0.

One potential advantage of tournament selection over all other forms is that itonly needs a preference ordering between pairs or groups of strings, and it can thuscope with situations where there is no formal objective function at all—in otherwords, it can deal with a purely subjective objective function!

However, tournament selection is also subject to arbitrary stochastic effects inthe same way as roulette-wheel selection—there is no guarantee that every stringwill appear in a given cycle. Indeed, using sampling with replacement there is aprobability of approximately e−1(≈ 0.368) that a given string will not appear at all.One way of coping with this, at the expense of a little extra computation, is to usea variance reduction technique from simulation theory. Saliby [110] distinguishesbetween the set effect and the sequence effect in drawing items from a finite pop-ulation. In applying his ideas here, we know that we need τ items to be drawn Mtimes, so we simply construct τ random permutations5 of the numbers 1, . . . ,M—theindices of the individuals in the population. These are concatenated into one longsequence which is then chopped up into M pieces, each containing the τ indicesof the individuals to be used in the consecutive tournaments. If M is not an exactmultiple of τ , there is the small chance of some distortion where the permutationsjoin, but this is a relatively minor problem.

5 There is a simple algorithm for doing this efficiently—see Nijenhuis and Wilf [111], for example,or look at the Stony Brook Algorithm Repository [112].

124 Colin R. Reeves

5.9 Crossover

Crossover is simply a matter of replacing some of the genes in one parent by thecorresponding genes of the other. Suppose we have two strings a and b, each con-sisting of six variables, i.e.,

(a1,a2,a3,a4,a5,a6) and (b1,b2,b3,b4,b5,b6),

which represent two possible solutions to a problem. One-point crossover (1X) hasbeen described earlier in the context of a binary alphabet. (Note that we have cho-sen here to leave the alphabet unspecified, to emphasize that binary representation isnot a critical aspect of GAs.) Two-point crossover (denoted by 2X) is very similar:two crosspoints are chosen at random from the numbers 1, . . . ,5, and a new solu-tion produced by combining the pieces of the original ‘parents’. For instance, if thecrosspoints were 2 and 4, the ‘offspring’ solutions would be

(a1,a2,b3,b4,a5,a6) and (b1,b2,a3,a4,b5,b6)

A similar prescription can be given for m-point crossover where m > 1.An early and thorough investigation of multipoint crossovers is that by Eshelman

et al. [113], who examined the biasing effect of traditional one-point crossover andconsidered a range of alternatives. Their central argument is that two sources ofbias exist to be exploited in a genetic algorithm: positional bias and distributionalbias. One-point crossover has considerable positional bias, in that it relies on thebuilding-block hypothesis, and if this is invalid, the bias may prevent the productionof good solutions.

On the other hand, 1X has no distributional bias, in that the crossover point ischosen randomly using the uniform distribution. But this lack of bias is not neces-sarily a good thing, as it limits the exchange of information between the parents.In [113], the possibilities of changing these biases, in particular by using multi-point crossover, were investigated and empirical evidence strongly supported thesuspicion that one-point crossover is not the best option. In fact, despite some am-biguities, the evidence seemed to point to an 8-point crossover operator as the bestoverall, in terms of the number of function evaluations needed to reach the globaloptimum, averaged over a range of problem types.

Another obvious alternative, which removes any bias, is to make the crossoverprocess completely random—the so-called uniform crossover. This can be seen mosteasily by observing that a crossover operator itself can be written as a binary stringor mask—in fact, when implementing crossover in a computer algorithm, this is theobvious way to do it. For example, the mask

1 1 0 0 1 1

represents the 2-point crossover used above, where a 1 means that the alleles aretaken from the first parent, while a 0 means they come from the second.

By generating the pattern of 0s and 1s stochastically (using a Bernoulli distribu-tion) we thus get uniform crossover (UX), which might generate a mask such as


1 0 1 0 0 1

which implies that the 1st, 3rd and 6th alleles are taken from the first parent, theothers from the second. This idea was first used by Syswerda [114], who implicitlyassumed the Bernoulli parameter p = 0.5. Of course, this is not necessary: we canbias UX towards one or the other parent by choosing p appropriately.

De Jong and Spears [115] produced a theoretical analysis that was able to char-acterize the amount of disruption introduced by a given crossover operator exactly.In particular, the amount of disruption in UX can be tuned by choosing differentvalues of p.

Of course, there are also many practical considerations that influence the imple-mentation of crossover. How often do we apply it? Some always do, others use astochastic approach, applying crossover with a probability χ < 1. Do we generateone offspring or two? In many cases there are natural ‘twin’ offspring resulting,but in more sophisticated problems it may be that only one offspring arises. Whenwe choose only one from two, how do we do it? In accordance with the stochasticnature of the GA, we may well decide to choose either of the offspring at random.Alternatively, we could bias the decision by making use of some other property suchas the fitness of the new individuals or the loss (or gain) in diversity that results inchoosing one rather than the other.

Booker [116] reported significant gains from using an adaptive crossover rate:the rate was varied according to a characteristic called percent involvement. Thisis simply the percentage of the current population that is producing offspring—toosmall a value is associated with loss of diversity and premature convergence.

5.9.1 Non-linear Crossover

In cases of non-linear encodings, crossover has to be reinterpreted. One of the mostfrequently occurring problems is where the solution space is the space of permu-tations (Πl) of the numbers 1, . . . , l—well-known examples of this include manyscheduling problems, and the famous travelling salesman problem (TSP).

For instance, the simple-minded application of 1X with crosspoint X = 2 in thefollowing case produces an infeasible solution:

P1 1 6 3 4 5 2 O1 1 6 1 2 6 5X

P2 4 3 1 2 6 5 O2 4 3 3 4 5 2

If this represents a TSP, the first offspring visits cities 1 and 6 twice, and never getsto cities 3 or 4. A moment’s thought is enough to realize that this type of behaviourwill be the rule, not an exception. Clearly we need to think of something rathersmarter if we are to be able to solve such problems.

One of the first ideas for such problems was the PMX (partially mappedcrossover) operator [94], which operates as follows: Two crossover points are cho-sen uniformly at random between 1 and l. The section between these points defines

126 Colin R. Reeves

an interchange mapping. Thus, in the example above, PMX (with crosspoints X=2and Y=5) might proceed as follows:

P1 1 6 3 4 5 2 O1 3 5 1 2 6 4X Y

P2 4 3 1 2 6 5 O2 2 1 3 4 5 6

Here the crossover points X and Y define an interchange mapping

3 ↔ 1 4 ↔ 2; 5 ↔ 6

on their respective strings, which means that the cut blocks have been swapped andnow appear in different contexts from before. Another possibility is to apply a binarymask, as in linear crossover, but with a different meaning. Such a mask, generatedas with UX, say, might be the following

1 0 1 0 0 1

which is applied to the parents in turn. First the components corresponding to 1s arecopied from one parent, and then those that correspond to 0s are taken in the orderthey appear from the second parent in order to fill the gaps. Thus the above examplegenerates the following pairs of strings:

P1 1 6 3 4 5 2 -> 1 _ 3 _ _ 2 O1 1 4 3 6 5 2

P2 4 3 1 2 6 5 -> 4 _ 1 _ _ 5 O2 4 6 1 3 2 5

5.10 Mutation

First we note that in the case when crossover-OR-mutation is used, we must firstdecide whether any mutation is carried out at all. Assuming that it is the concept ofmutation is even simpler than crossover, and again, this can easily be represented asa bit-string, so we generate a mask such as

0 1 0 0 0 1

using a Bernoulli distribution at each locus—with a small value of p in this case.(The above example would then imply that the 2nd and 6th genes are assigned newallele values.) However, it appears that there are variant ways of implementing thissimple idea that can make a substantial difference to the performance of a GA.The naive idea would be to draw a random number for every gene in the stringand compare it to μ , but this is potentially expensive in terms of computation ifthe strings are long and the population is large. An efficient alternative is to draw arandom variate from a Poisson distribution with parameter λ , where λ is the averagenumber of mutations per chromosome. A common value for λ is 1—in other words,if l is the string length, the (bit-wise) mutation rate is μ = 1/l, which as early as1966 [118] was shown to be in some sense an ‘optimal’ mutation rate. If our Poisson


random draw proposes that there are (say) m mutations, we draw m random numbers(without replacement) uniformly distributed between 1 and l in order to specify theloci where mutation is to take place.

In the case of binary strings, mutation simply means complementing the chosenbit(s). More generally, when there are several possible allele values for each gene,if we decide to change a particular allele, we must provide some means of decidingwhat its new value should be. This could be a random choice, but if (as in somecases) there is some ordinal relation between allele values, it may be more sensibleto restrict the choice to alleles that are close to the current value or at least to biasthe probability distribution in their favour.

It is often suggested that mutation has a somewhat secondary function, that ofhelping to preserve a reasonable level of population diversity—an insurance policywhich enables the process to escape from sub-optimal regions of the solution space,but not all authors agree. Proponents of evolutionary programming ([119], for ex-ample), consider crossover to be an irrelevance, and mutation plays the major role.The balance between crossover and mutation is often a problem-specific one, anddefinite guidelines are hard to give.

However, several authors have suggested some type of adaptive mutation: forexample, Fogarty [120] experimented with different mutation rates at different loci.Reeves [100] varied the mutation probability according to the diversity in the popu-lation (measured in terms of the coefficient of variation of fitnesses). More sophis-ticated procedures are possible, and anecdotal evidence suggests that many authorsuse some sort of diversity maintenance policy. In this connection, it should also bementioned that there is interest currently in ‘parameter-less’ GAs. It is impossibleto eliminate all parameter values, of course, but there has always been interest insome sort of adaptation as the search proceeds, not only for mutation rates but alsofor other parameters, such as population size. Eiben et al. [121] summarize some ofthe recent work in this area.

Finally, it should be no surprise that the values of different parameters interactwith each other, in terms of the overall performance of the GA. For example, choos-ing a high selection pressure may mean that we also need a high mutation rate inorder to avoid premature convergence. De Jong [43] has an extensive discussion onsuch matters.

5.11 New Population

Holland’s original GA assumed a generational approach: selection, recombinationand mutation were applied to a population of M chromosomes until a new set of Mindividuals had been generated. This set then became the new population. From anoptimization viewpoint this seems an odd thing to do—we may have spent consid-erable effort obtaining a good solution, only to run the risk of throwing it awayand thus preventing it from taking part in further reproduction. For this reason,De Jong [5] introduced the concepts of elitism and population overlaps. His ideas

128 Colin R. Reeves

are simple—an elitist strategy ensures the survival of the best individual so far bypreserving it and replacing only the remaining (M− 1) members of the populationwith new strings. Overlapping populations take this a stage further by replacing onlya fraction G (the generation gap) of the population at each generation. Finally, tak-ing this to its logical conclusion produces the so-called steady-state or incrementalstrategies, in which only one new chromosome (or sometimes a pair) is generated ateach stage. Davis [32] gives a good general introduction to this type of GA.

Slightly different strategies are commonly used in the ES community, which tra-ditionally designates them either λ ,μ or λ + μ . In the first case, μ(> λ ) offspringare generated from λ parents, and the best λ of these offspring are chosen to startthe next generation. For the + strategy, μ (not necessarily > λ ) offspring are gen-erated and the best λ individuals are chosen from the combined set of parents andoffspring.

In the case of incremental reproduction it is also necessary to select membersof the population for deletion. Some GAs have assumed that parents are replacedby their children. Many implementations, such as Whitley’s GENITOR [109], usethe tactic of deleting the worst member(s) of the population, although (as Goldbergand Deb [122] have pointed out) this exerts a very strong selective pressure on thesearch, which may need fairly large populations and high mutation rates to preventa rapid loss of diversity. A milder prescription is to select from the worst p% of thepopulation (for example, Reeves [100] used p = 50, i.e., selection from those worsethan the median). This is easily implemented when rank-based selection is used. Yetanother approach is to base deletion on the age of the strings.

5.11.1 Diversity Maintenance

As hinted above, one of the keys to good performance (in nature as well as in GAs) isto maintain the diversity of the population as long as possible. The effect of selectionis to reduce diversity, and some methods can reduce diversity very quickly. This canbe mitigated by having larger populations or by having greater mutation rates, butother techniques are also often employed.

A popular approach, commonly linked with steady-state or incremental GAs, isto use a ‘no-duplicates’ policy [32]. This means that the offspring are not allowedinto the population if they are merely clones of existing individuals. The downside,of course, is the need to compare each current individual with the new candidate,which adds to the computational effort needed—an important consideration withlarge populations. (In principle, some sort of ‘hashing’ approach could be used tospeed this process up, but whether this has ever been tried is not clear.)

We can of course take steps to reduce the chance of cloning before offspring aregenerated. For instance, with 1X, the two strings

1 1 0 1 0 0 11 1 0 0 0 1 0


will generate only clones if the crossover point is any of the first three positions.Booker [116] suggested that before applying crossover, we should examine theselected parents to find suitable crossover points. This entails computing an‘exclusive-OR’ (XOR) between the parents, so that only positions between theoutermost 1s of the XOR string (the ‘reduced surrogate’) should be considered ascrossover points. Thus in the example above, the XOR string is

0 0 0 1 0 1 1

so that, as previously stated, only the last three crossover points will give rise to adifferent string.

5.12 Representation

As remarked in Section 5.1, the focus in this handbook is on using GAs as optimizersin a search space, given a suitable encoding and fitness function. We now considerhow the search space S might be constructed in some generic cases.

5.12.1 Binary Problems

In some problems a binary encoding might arise naturally. Consider the operationalresearch problem known as the knapsack problem, stated as follows.

Example 1 (The 0-1 knapsack problem) A set of n items is available to be packedinto a knapsack with capacity C units. Item i has value vi and uses up ci units of ca-pacity. Determine the subset I of items which should be packed in order to maximize

∑i∈I

vi

such that

∑i∈I

ci ≤C.

If we define

xi ={

1 if item i is packed0 otherwise

the knapsack problem can be re-formulated as an integer program:

maximizen

∑i=1

xivi,

such thatn

∑i=1

xici ≤C,

130 Colin R. Reeves

from which it is clear that we can define a solution as a binary string of length n.In this case there is thus no distinction between genotype and phenotype.

However, such problems are not necessarily easy to solve with a GA. In this case,the presence of constraints is likely to cause difficulties—two feasible parents maynot produce feasible offspring, unless special crossover operators are constructed.In fact, such problems as these are really subset selection problems, which are besttackled by other means [123], despite the seductiveness of the binary encoding.It is now widely recognized that ‘natural’ binary encodings nearly always bringsubstantial problems for simple GAs.

5.12.2 Discrete (but Not Binary) Problems

There are cases in which a discrete alphabet of higher cardinality than 2 mightbe appropriate. The rotor-stacking problem, as originally described by McKee andReed [124], is a good example.

Example 2 A set of n rotors are available, each of which has k holes drilled init. The rotors have to be assembled into a unit by stacking them and bolting themtogether, as in Figure 5.4. Because the rotors are not perfectly flat, stacking them indifferent orientations will lead to assemblies with different characteristics in termsof deviations from true symmetry, with the consequent effect (in operation) that theassembled unit will wobble as it spins. The objective is to find which of all thepossible combinations of orientations produce the least deviation.

··

·

Fig. 5.4 Rotor-stacking problem with n = 5 rotors and k = 3 holes.

In this case a k-ary coding is natural. A solution is represented by a string oflength n, each gene corresponding to a rotor and the alleles, drawn from {1, . . . ,k},representing the orientation (relative to a fixed datum) of the holes. Thus, the string(1322) represents a solution to a 4-rotor problem where hole 1 of the first rotor isaligned with hole 3 of the second and hole 2 of the third and fourth. Of course, itwould be possible to encode the alleles as binary strings, but there seems little pointin so doing—particularly if k is not a power of 2, as there will then be some binarystrings that do not correspond to any actual orientation.

This seems very straightforward, although there is a subtle point that could beoverlooked. The assignment of labels to the holes is arbitrary, and this creates the


problem of ‘competing conventions’ as it has been called6. For example, given anatural order for labelling each rotor, the string (3211) represents the same solutionas (1322). This can be alleviated in this case by fixing the labelling for one rotor, sothat a solution can be encoded by a string of length (n−1).

As far as the operators are concerned, standard crossovers can be used here, butmutation needs some careful consideration in the case of k-ary coding, as outlinedin Section 5.10.

5.12.3 Permutation Problems

There are also some problems where the ‘obvious’ choice of representation isdefined, not over a set, but over a permutation. The TSP is one of many prob-lems for which this is true. As another example, consider the permutation flowshopsequencing problem (PFSP).

Example 3 Suppose we have n jobs to be processed on m machines, where theprocessing time for job i on machine j is given by p(i, j). For a job permutation{π1,π2, . . . ,πn}, we calculate the completion times C(πi, j) as follows:

C(π1,1) = p(π1,1)C(πi,1) = C(πi−1,1)+ p(πi,1) for i = 2, . . . ,n

C(π1, j) = C(π1, j−1)+ p(π1, j) for j = 2, . . . ,m

C(πi, j) = max{C(πi−1, j),C(πi, j−1)}+ p(πi, j)for i = 2, . . . ,n; j = 2, . . . ,m

The PFSP is then to find a permutation π∗ in the set of all permutations Π such that

f (π∗)≤ f (π) ∀π ∈ Π .

(Several performance measures f (·) are possible; common ones are the maximumor mean completion time.)

Here the natural encoding (although not the only one) is simply the permutationof the jobs as used to calculate the completion times. So the solution (1462537), forexample, simply means that job 1 is first on each machine, then job 4, job 6, etc.

Unfortunately, the standard crossover operators patently fail to preserve the per-mutation except in very fortunate circumstances, as discussed in Section 5.9.1. Somesolutions to this problem were outlined there; more comprehensive discussion ofpossible methods of attack is contained in [126, 127], while [100, 128] describesome approaches of particular relevance to the PFSP.

6 This phenomenon is a common one whenever the coding function c(·) is not injective. It hasbeen observed in problems ranging from optimizing neural nets to the TSP. Radcliffe, who calls it‘degeneracy’ [125], has presented the most thorough analysis of this problem and how to treat it.

132 Colin R. Reeves

5.12.4 Non-binary Problems

In many cases the natural variables for the problem are not binary, but integer orreal-valued. In such cases a transformation to a binary string is required first. (Notethat this is a different situation from the rotor-stacking example, where the inte-gers were merely labels: here the values are assumed to be meaningful as numbers.)While the main thrust of metaheuristics research and application is directed to dis-crete optimization, it is perhaps appropriate to mention these other problems here.

Example 4 It is required to maximize

f (x) = x3 −60x2 +900x+100

over the search space X = {x : x ∈ ZZ;x ∈ {0,31}}, i.e., the solution x∗ is requiredto be an integer in the range [0,31].

To use the conventional form of genetic algorithm here, we would use a stringof 5 binary digits with the standard binary to integer mapping, i.e., (0,0,0,0,0) =0, . . . ,(1,1,1,1,1) = 31. Of course, in practice we could solve such a problem eas-ily without recourse to encoding the decision variable in this way, but it illustratesneatly the sort of optimization problem to which GAs are often applied. Such prob-lems assume first that we know the domain of each of our decision variables, andsecond that we have some idea of the precision with which we need to specify oureventual solution. Given these two ingredients, we can determine the number of bitsneeded for each decision variable and concatenate them to form the chromosome.More information on this topic can be found in [12].

5.13 Random Numbers

As GAs are stochastic in nature, it is clear that a reliable random number source isvery important. Most computer systems have built-in rand() functions, and that isthe usual method of generating random numbers. Not all random number generatorsare reliable, however, as Ross [129] has pointed out, and it is a good idea to use onethat has been thoroughly tested, such as those described in the Numerical Recipesseries [130].

5.14 Conclusions

While this exposition has covered the basic principles of GAs, the number ofvariations that have been suggested is enormous. Probably everybody’s GA isunique! Many variations in population size, in initialization methods, in fitnessdefinition, in selection and replacement strategies, in crossover and mutation are


obviously possible. Some have added information such as age, or artificial tags, tochromosomes; others have allowed varying population sizes or induced the forma-tion of multiple populations in ‘niches’. It is in the nature of GAs that parallel pro-cessing can often be used to advantage, and here again, there are many possibilities,ranging from simple parallelization of function evaluations to very sophisticatedimplementations that add a spatial aspect to the algorithm.

The GA community has yet to reach a consensus on any of these things, andin the light of the NFLT, this is perhaps not surprising. However, some ideas doemerge as a reasonable set of recommendations. From a practitioner’s viewpoint,Levine made the following observations:

1. A steady-state (or incremental) approach is generally more effective and effi-cient than a generational method.

2. Don’t use simple roulette-wheel selection. Tournament selection or SUS isbetter.

3. Don’t use one-point crossover. UX or 2X should be preferred.4. Make use of an adaptive mutation rate—one that is fixed throughout the search

(even at 1/l) is too inflexible.5. Hybridize wherever possible; don’t use a GA as a black box, but make use of

any problem-specific information that you have.

Not everyone will agree with this particular list, and there is a conflict inher-ent in the first two points, since SUS functions best in a generational setting.Broadly speaking, however, it is one with which many researchers would becomfortable. Two other points could be added:

6. Make diversity maintenance a priority.7. Don’t be afraid to run the GA several times.

Why this last point? Statements are frequently made that GAs can find globaloptima. Well, they can—but usually they tend to converge to some other ‘attractor’.In fact, there is some evidence [131] that even with very large populations the at-tractors are a subset of the local optima relating to a neighbourhood search. Withpractical population sizes, the attractors may not even be restricted to such a set andmay be some distance from global optimality. It thus makes sense to explore severalalternatives.

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Chapter 5 Genetic Algorithmshomes.di.unimi.it/.../Didattica/AlgoritmiEuristici/MaterialeAE/Ch05-GA… · Genetic Algorithms Colin R. Reeves Abstract Genetic algorithms (GAs) have